On stability of Abrikosov vortex lattices

TL;DR

This paper investigates the stability of Abrikosov vortex lattices in superconductors using Ginzburg-Landau equations, establishing conditions under which these lattices are stable or unstable depending on the superconductor type and lattice shape.

Contribution

It provides a stability analysis of Abrikosov vortex lattices for the Gorkov-Eliashberg-Schmid equations, considering arbitrary lattice shapes and near the second critical magnetic field.

Findings

Vortex lattices are stable in Type II superconductors when certain functions are positive.

Lattices are unstable in Type I superconductors or when the functions are negative.

Stability depends on the superconductor type and lattice shape.

Abstract

The Ginzburg-Landau equations play a key role in superconductivity and particle physics. They inspired many imitations in other areas of physics. These equations have two remarkable classes of solutions -- vortices and (Abrikosov) vortex lattices. For the standard cylindrical geometry, the existence theory for these solutions, as well as the stability theory of vortices are well developed. The latter is done within the context of the time-dependent Ginzburg-Landau equations -- the Gorkov-Eliashberg-Schmid equations of superconductivity -- and the abelian Higgs model of particle physics. We study stability of Abrikosov vortex lattices under finite energy perturbations satisfying a natural parity condition (both defined precisely in the text) for the dynamics given by the Gorkov-Eliashberg-Schmid equations. For magnetic fields close to the second critical magnetic field and for arbitrary lattice shapes, we prove that there exist two functions on the space of lattices, such that Abrikosov vortex lattice solutions are asymptotically stable, provided the superconductor is of Type II and these functions are positive, and unstable, for superconductors of Type I, or if one of these functions is negative.